KR20220170751A - Cordic computation of sin/cos using combined approach in associative memory - Google Patents

Abstract

A method for an associative memory device comprises the steps of providing a look up table (LUT) with all possible solutions for N first iterations of a CORDIC algorithm; receiving a plurality of input angles; concurrently computing a location index for each of the plurality of input angles, and concurrently storing each index in a column of the associative memory device; copying a solution from the LUT in the location index to a plurality of columns associated with the index; and concurrently performing M additional iterations of the CORDIC algorithm in the plurality of columns to compute the value of a trigonometrical function for each angle.

Description

CORDIC COMPUTATION OF SIN/COS USING COMBINED APPROACH IN ASSOCIATIVE MEMORY}

CROSS REFERENCES TO RELATED APPLICATIONS

This application claims priority from U.S. Provisional Application Serial No. 63/213,780, filed on June 23, 2021, which is incorporated herein by reference.

The present invention relates generally to computing trigonometric functions, and more particularly to the use of CORDIC algorithms within associative memory devices.

The CORDIC (COordinate Rotation Digital Computer) algorithm is an algorithm for calculating trigonometric functions such as sine and cosine. It is used like an approximation function in all popular graphing calculators.

The CORDIC algorithm has developed around the idea of "rotating" the phase of a complex number by multiplying it by a series of constant values. The multiplication may be by a power of two, which may be implemented in hardware using transitions and additions.

Figure 1A, to which reference is now made, illustrates a point (x, y) on a unit circle (a circle with a radius equal to 1). The point (x, y) can be expressed as two trigonometric functions of angle α: x = cos(α) and y = sin(α).

를 감소시킴으로써 시계방향 및/또는 반시계방향 회전의 연속적인 반복을 수행한다. 반복은 각도 0에서 시작할 수 있고, 여기에서 알고리즘의 각각의 단계는 각도 α를 향한 회전을 수행한다. 또는, 반복은 각도 α에서 시작할 수 있고, 여기에서 알고리즘의 각각의 단계는 각도 0을 향한 회전을 수행한다.The CORDIC algorithm is angular

Perform successive iterations of clockwise and/or counterclockwise rotation by decreasing . The iteration can start at angle 0, where each step of the algorithm performs a rotation towards angle α. Or, the iteration can start at angle α, where each step of the algorithm performs a rotation towards angle zero.

)은 반시계방향으로 회전하여 단위 원의 포인트 v₁에 도달할 수 있다. 이러한 예에서 각도 α는

보다 크다; 그러므로, 다음 반복 시에, 회전각(

)도 역시 반시계방향이고 각도 α에 더 가깝게 도착하여 포인트 v₂에 도달한다. 결과적으로 얻어지는 각도(

+

)는 각도 α보다 크고, 따라서 다음 반복 시에, 회전각(

)은 시계방향으로 회전하여 각도 α에 더 가까운 각도(

)가 될 수 있다. 각도

가 α에 더 가까워지게 하는 회전 프로시저는 미리 결정된(T) 회수의 반복을 계속한다. 결과적으로 얻어지는 각도

및 α 사이의 차이는 이러한 프로시저에서의 최대의 오차(mistake)를 나타내고, 이것은 1/2^T이다.FIG. 1B, to which reference is now made, is an example of a CORDIC algorithm used iteratively to determine the value of the sine or cosine of angle α, starting at angle zero. In the first iteration, the first rotation angle (

) can rotate counterclockwise to reach the point v ₁ on the unit circle. In this example the angle α is

greater than; Therefore, in the next iteration, the rotation angle (

) is also counterclockwise and arrives closer to angle α to point v ₂ . The resulting angle (

+

) is greater than the angle α, so in the next iteration, the rotation angle (

) rotates clockwise to an angle closer to angle α (

) can be Angle

The rotation procedure, which brings x closer to α, continues for a predetermined (T) number of iterations. the resulting angle

The difference between α and α represents the maximum misstake in this procedure, which is 1/2 ^T.

로서 표현되는 경우, 모든 반복은 회전을 연산하고, 이것은 벡터

를 수학식 1에 표현된 각도

를 나타내는 회전 매트릭스

로 승산함으로써 수행된다:point(x, y) is a vector

, every iteration computes a rotation, which is a vector

the angle expressed in Equation 1

Rotation matrix representing

This is done by multiplying by:

는 수학식 2에 의해서 규정된다:rotation matrix

is defined by Equation 2:

이 되도록 회전각

를 선택하면, 탄젠트로 승산하는 것을 수학식 3에 표현된 것처럼 2의 거듭제곱으로 나누는 것으로 대체할 수 있다(그리고 위에서 언급된 바와 같이, 2의 거듭제곱으로 나누는 것은 HW에서 천이 동작에 의하여 구현될 수 있다).

Rotation angle so that

, we can replace multiplication by tangent with division by a power of 2 as expressed in Equation 3 (and as mentioned above, division by a power of 2 can be implemented by a transition operation in HW). can).

의 값은, 각각의

에 대해서 사전에 계산될 수 있다expressed by Equation 4

The value of each

can be calculated in advance for

에서의

의 값은 회전 방향을 결정하고, 여기에서 +1은 반시계방향 회전이고 -1은 시계방향 회전이다.

in

The value of determines the rotation direction, where +1 is counterclockwise rotation and -1 is clockwise rotation.

The resulting equations used to compute cosine and sine values using a sequence of iterations of the CORDIC algorithm can be expressed by equations 5 and 6, respectively. The value x is the cosine, and the value y is the sine.

As described herein above, the CORDIC algorithm includes a set of iterations, where at each step an angle of known magnitude is added or subtracted from the temporary angle calculated in the previous step, cosine and sine values (x and y value) to reach the size of the required angle α.

는 CORDIC 알고리즘의

회의 반복을 수행한 이후에 각도 α에 대한 사인 및 코사인 값을 제공한다.

회의 반복 이후에 CORDIC 알고리즘의 결과를 저장하는 테이블의 크기는 T=

이다.Alternatively (instead of performing the actual computation), all possible results of the CORDIC algorithm iterated T times may be pre-stored in a look-up table (LUT). Each entry in the LUT

is the CORDIC algorithm

After performing round iterations, we give the sine and cosine values for angle α.

After meeting iterations, the size of the table storing the results of the CORDIC algorithm is T=

to be.

It can be understood that the number of rotations T determines the accuracy of the result, and that increasing the number of rotations increases the accuracy of the result (sine/cosine value).

When using a LUT, the number of iterations T determines the size of the table and the number of bits in the index pointing to the table. When performing actual calculations, the number of iterations T determines the computational complexity.

According to a preferred embodiment of the present invention, a method for an associative memory device is provided. The method includes providing a look-up table (LUT) having all possible solutions for the first N iterations of the CORDIC algorithm, receiving a plurality of input angles, a position of the plurality of input angles for each input angle concurrently computing indices, and concurrently storing each index into a row of the associative memory device. The method includes copying a solution from the LUT at the position index into a plurality of columns associated with the index, and calculating a value of a trigonometric function for each of the plurality of input angles, and concurrently performing M additional iterations of the CORDIC algorithm in the column.

Furthermore, according to a preferred embodiment of the present invention, calculating the position index simultaneously is performed for each of the input angles simultaneously, and the following steps: initializing the temporary angle to the value of the input angle, bit determining a value and sign based on a comparison between the temporary angle and zero, assigning the bit to the position index, calculating a rotation angle based on the sign and a predetermined angle, and the rotation angle and adding to the temporary angle. The steps of initializing, determining, assigning, and calculating are repeated N times and result in an N-bit position index.

Moreover, according to a preferred embodiment of the present invention, N = 5 and M = 5.

Further, according to a preferred embodiment of the present invention, the step of copying the solution proceeds sequentially through all entries of the LUT, and converts the X values and Y values from the position indexes in the LUT to the same computed position index. Branching involves copying to all columns simultaneously.

According to a preferred embodiment of the present invention, sine and cosine estimators are provided. The sine and cosine estimator includes a look-up table (LUT) for storing previously calculated sine and cosine values for the first N iterations of the CORDIC algorithm, an associative memory array for storing information related to a plurality of angles, and the plurality of A LUT index builder for simultaneously building an index reflecting a rotation of a predefined rotation angle for each of the angular indices, a LUT value for assigning a value from an entry in the LUT to a column of the associative memory array that shares the same index. an allocator, and simultaneously computing M additional iterations of the CORDIC algorithm on a plurality of columns of the associative memory array, thereby providing sine and cosine values after N + M iterations of the CORDIC algorithm to the plurality of angles. Includes CORDIC computers.

Furthermore, according to a preferred embodiment of the present invention, the LUT contains 2 ⁵ entries, N = 5.

The subject matter which is regarded as the present invention is particularly pointed out and distinctly claimed in the concluding section of this specification. However, the present invention, both as an organization and a method of operation, together with its objects, features and advantages, will be best understood by reference to the specific details for carrying out the invention that follow when read in conjunction with the accompanying drawings:
1A is a schematic illustration of a point (x, y) on a unit circle;
Figure lb is a schematic illustration of a state-of-the-art CORDIC algorithm used iteratively to determine the value of the sine or cosine of angle α;
Figure 2 is an exemplary diagram of a lookup table (LUT) storing all possible results of the CORDIC algorithm after N iterations;
3 is a schematic diagram of a flow operating in accordance with one embodiment of the present invention, describing a unified approach for computing the sine and cosine of angle α;
4 is a schematic diagram of a sine/cos estimator constructed and operative in accordance with one embodiment of the present invention;
Fig. 5 is a schematic diagram of a flow according to one embodiment of the present invention, performed by the sine/cos estimator of Fig. 4, for generating an angular index pointing to a LUT for each input;
Figure 6 is a schematic diagram of a flow according to one embodiment of the present invention, performed by the sine/cos estimator of Figure 4, for copying the values of a LUT into a column of an associative memory array;
Fig. 7 is a schematic diagram of a flow according to one embodiment of the present invention, performed by the sine/cos estimator of Fig. 4, for computing additional iterations of the CORDIC algorithm; And
8 is a schematic diagram of an associative memory array used by the FIG. 4 sine/cos estimator and constructed and operative in accordance with one embodiment of the present invention.
For purposes of concise and clear explanation, it will be understood that elements shown in the drawings are not necessarily drawn to scale. For example, the dimensions of some of the elements may be clearly displayed exaggerated relative to other elements. Furthermore, where appropriate, reference numbers may be used repeatedly in the drawings to indicate corresponding or similar elements.

In the detailed description that follows, numerous specific details are set forth in order to provide a thorough understanding of the present invention. However, those skilled in the art will understand that the invention may be practiced without these specific details. In other instances, well-known methods, procedures and components have not been described in detail so as not to unnecessarily obscure aspects of the invention.

Many applications, such as synthetic aperture radar (SAR) algorithms used to generate images from radar pulses, require efficient simultaneous computation of trigonometric functions of multiple angles. Applicant claims U.S. Patent No. 9,558,812 (entitled "SRAM multi-cell operations") and U.S. Patent No. 10,832,746 (entitled "Non- It has been found that associative memory devices such as those described in “volatile in-memory computing devices” can simultaneously and efficiently compute sine and cosine values for multiple angles.

Applicants have found that implementing the CORDIC algorithm within such a device can provide simultaneous trigonometric function calculations with constant complexity. The complexity depends only on the total number of iterations (T) performed by the CORDIC algorithm and will not depend on the number of input angles for which sine or cosine values are required (this can be a large number, for example 32K angles). can

Applicant argues that a unified approach in which the calculations specified by the CORDIC algorithm are performed for additional M iterations after the results of the first N iterations of the CORDIC algorithm are stored in the LUT, either use the LUT for all T iterations or CORDIC We also find that we can improve the performance of the total computation (T = M + N iterations) compared to performing T iterations of the algorithm.

0.00097656203 일 수 있다.In one embodiment of the unified approach, the LUT may pre-store all possible values that can be obtained after 5 iterations of the CORDIC algorithm, and the next 5 iterations of the CORDIC algorithm will be simultaneously obtained in the associative memory array. can The accuracy of the result after 10 iterations of the CORDIC algorithm is

It can be 0.00097656203.

<N에 대하여

이 성립하도록 선택될 수 있는, N 개의 미리 규정되고 감소되는 크기를 가지는 각도

를 사용한 N 회의 반복 이후의 CORDIC 알고리즘의 모든 가능한 결과를 저장하는 LUT(20)의 일 예이다.2, to which reference will now be made, each 0<

About <N

angles having N predefined and decreasing magnitudes, which can be selected such that

An example of a LUT 20 that stores all possible results of the CORDIC algorithm after N iterations using

가 반복

에서의 회전각

의 방향을 표시하도록 구축될 수 있다. 인덱스의 LSB의 값은 첫 번째 회전의 방향을 표시하고, 다음 비트는 다음 회전의 방향을 표시하는 식인데, 이것은 마지막 회전의 방향을 표시하는 MSB에 도달할 때까지 마찬가지이다. 인덱스 내의 비트

에 대한 값 0은

만큼의 반시계방향 회전을 표시하고, 인덱스 내의 비트

에 대한 값 1은

만큼의 시계방향 회전을 표시한다.Column 21 of LUT 20 may provide an index pointing to the LUT. The index pointing to the LUT is each bit of the index

repeat

angle of rotation at

It can be built to indicate the direction of The value of the LSB of the index represents the direction of the first rotation, the next bit represents the direction of the next rotation, and so on, until the MSB representing the direction of the last rotation is reached. bits in index

A value of 0 for

indicates a counterclockwise rotation by the number of bits in the index

A value of 1 for

indicates clockwise rotation.

)의 값을 제공할 수 있고, 열(23)은 각각의 가능한 인덱스에 대한 CORDIC 알고리즘의 5 회의 반복 이후의 Y(sin(

)의 값을 제공할 수 있다.Column 22 of LUT(20) is X(cos(

), column 23 is Y(sin() after 5 iterations of the CORDIC algorithm for each possible index.

) can be provided.

의 연속 회전에 의하여 유도되는 각도

의 사인(Y) 및 코사인(X) 값을 제공할 수 있다.Each row 24 in the LUT 20 has a predefined angle

angle induced by successive rotations of

can provide the sine (Y) and cosine (X) values of

에 대한 CORDIC 알고리즘의 처음 N 회의 반복을 대체하는 2^N 개의 LUT 룩업, 및 모든 각

에 대한 CORDIC 알고리즘의 다음 M 회의 반복의 병렬 계산은 T 회의 회전(T = N + M 개의 회전각)의 전체 범위를 커버한다.3, to which reference is now made, is a schematic diagram of a flow 300 operating in accordance with one embodiment of the present invention, describing an integrated approach. In the unified approach, the sine and cosine calculations for multiple angles can be integrated from parallel computations of the CORDIC algorithm for 2 ^N LUT lookups and the next M iterations. multiple angles

2 ^N LUT lookups replacing the first N iterations of the CORDIC algorithm for , and every

The parallel computation of the next M iterations of the CORDIC algorithm for , covers the full range of T rotations (T = N + M rotation angles).

를 수신할 수 있다.At step 320, flow 300 directs a number of angles for which cosine and/or sine values are required.

can receive

에 대하여, LUT를 가리키는 인덱스를 동시에 계산할 수 있고, LUT 값으로부터 CORDIC 알고리즘의 처음 N 회의 반복에 대해서 사전에 계산된 각각의 각도

에 대한 값들

및

를 독출할 수 있다.At step 340, flow 300 may perform a first part of a unified approach using a LUT. At step 340, flow 300 is directed to each angle

For , the index pointing to the LUT can be calculated simultaneously, and each angle calculated in advance for the first N iterations of the CORDIC algorithm from the LUT value

values for

and

can be read.

에 대하여, 단계(340)에 의해서 제공된 값들

및

로부터 시작하여 CORDIC 알고리즘의 다음 M 회의 반복을 동시에 계산하고 총 T 회의 반복 이후에

및

의 최종 값들을 생성할 수 있다.At step 360, flow 300 may perform a second portion of the unified approach and may perform M additional iterations of the CORDIC computation. At step 360, flow 300 is directed to each angle

For , the values provided by step 340

and

Simultaneously compute the next M iterations of the CORDIC algorithm starting from, and after a total of T iterations

and

can produce the final values of

및

를 제공할 수 있다.At step 380, flow 300 returns the final value, which is the estimated value of the cosine and sine trigonometric functions after T = N + M iterations.

and

can provide.

4, to which reference is now made, is a schematic illustration of a sine/cos estimator 400 constructed and operative in accordance with one embodiment of the present invention. Sine/cos estimator 400 may use a unified approach and may implement flow 300 of FIG. 3 . The sine/cos estimator 400 includes an associative memory array 410, a LUT 420, a LUT index builder 430, a LUT value assigner 440, and a CORDIC computer 450.

에 관련된 모든 중간 계산 결과가 연상 메모리 어레이(410)의 열 k에 저장될 수 있다.The associative memory array 410 may include a plurality of cells 411 arranged in a matrix having bit lines 413 (columns) and word lines 415 (rows). All cells 411 in the same column can be connected to the same bit line 413, and all cells 411 in the same row can be connected to the same word line 415. Associative memory array 410 is described in detail in FIG. 7 discussed later herein. Angle

All intermediate calculation results related to may be stored in column k of the associative memory array 410 .

LUT 420 may be an embodiment of LUT 20 (FIG. 2) that stores the result of the CORDIC algorithm after N (eg, 5) rotations. In such an embodiment, the number of bits in the index pointing to LUT 420 may be 5, and the size of LUT 420 may be 2 ⁵ = 32. In embodiments of the invention, it can be appreciated that the size of the LUT 420 (32 entries) can be optimized to match the current performance of the hardware used to build the sine/cos estimator 400. The size of the LUT 420 is not limited to the size indicated above and can be changed to obtain the best performance as hardware performance improves.

에 대해서, LUT(420)를 가리키는 인덱스로서 사용될 인덱스 J_k를 동시에 생성할 수 있다. LUT 인덱스 빌더(430)는 생성된 인덱스 J_k를 연상 메모리 어레이(410)의 각도

에 관련된 그러한 비트 라인(413)에 기록할 수 있다. 각각의 각도

에 대한 인덱스 J_k를 생성하기 위한 흐름은 본 명세서에서 후술되는 도 5에서 설명된다.The LUT index builder 430, constructed and operated according to an embodiment of the present invention,

For , an index J _k to be used as an index pointing to the LUT 420 can be generated at the same time. The LUT index builder 430 is an angle of the memory array 410 associated with the generated index J _k .

You can write to those bit lines 413 related to . each angle

A flow for generating an index J _k for is described in FIG. 5 described later in this specification.

및

를 인덱스 J_k와 연관된 연상 메모리 어레이(410)의 그러한 열에 동시에 기록할 수 있다. LUT 값 할당기(440)에 의해 수행되는 흐름은 본 명세서에서 후술되는 도 6에서 설명된다.A LUT value allocator 440 constructed and operating in accordance with one embodiment of the present invention can proceed through all entries in the LUT 420, and a value for each index J _k

and

can be simultaneously written to those columns of the associative memory array 410 associated with index J _k . The flow performed by LUT value assigner 440 is illustrated in FIG. 6 described later herein.

에 대해서, 연상 메모리 어레이(410)의 열 내에 저장된 값

및

로부터 시작하여 CORDIC 알고리즘의 M 회의 반복을 동시에 계산할 수 있다. CORDIC 컴퓨터(450)에 의해 수행되는 흐름은 본 명세서에서 후술되는 도 7에서 설명된다.A CORDIC computer 450 constructed and operating in accordance with one embodiment of the present invention is

For , the value stored in the column of the associative memory array 410

and

Starting from , M iterations of the CORDIC algorithm can be computed simultaneously. The flow performed by CORDIC computer 450 is illustrated in FIG. 7 discussed later herein.

에 대한 인덱스를 계산하기 위한 흐름(500)의 개략도이다. 각각의 입력 각도 α_k에 대한 LUT(420)를 가리키는 인덱스는 해당 인덱스 내의 각각의 비트에 대하여 반복적으로 계산될 수 있다.5, to which reference is now made, each angle, implemented in accordance with one embodiment of the present invention and performed by the LUT index builder 430.

A schematic diagram of a flow 500 for calculating an index for . An index pointing to the LUT 420 for each input angle α _k may be iteratively calculated for each bit within that index.

를 수신할 수 있다.In step 501, the LUT index builder 430 selects a plurality of angles for which sine or cosine values are required.

can receive

를 입력 각도 α_k의 값으로 초기화하고, 반복자(iterator)

(해당 인덱스의 비트들에 걸쳐서 반복하기 위하여 사용됨)를 0으로 초기화할 수 있다.At step 510, the LUT index builder 430

is initialized to the value of the input angle α _k , and an iterator

(used to iterate over the bits of that index) can be initialized to 0.

에 대해서 동시에 시작할 수 있다. 단계들(530, 543, 546, 550 및 560)에서, LUT 인덱스 빌더(430)는 회전을 수행할 수 있고, LUT(420)를 가리키는 복수 개의 인덱스 J_k (인덱스는 각각의 각도 α_k에 관련됨) 내의 비트들의 값을 결정할 수 있다. LUT 인덱스 빌더(430)는 각각의 인덱스 J_k에 대하여 N 개의 비트를 계산하기 위하여 N 회의 회전을 수행할 수 있다. 각각의 반복

에서, 인덱스 J_k의 비트

는 임시 각도

의 값을 0과 비교하고 관련된 값(0 또는 1)을 J_k의 비트

에 할당함으로써 결정될 수 있다.At step 520, the LUT index builder 430 converts the first iteration to all angles.

can start at the same time. At steps 530, 543, 546, 550 and 560, the LUT index builder 430 may perform a rotation, and a plurality of indices J _k pointing to the LUT 420 (an index associated with each angle α _k ). ) It is possible to determine the values of the bits in. The LUT index builder 430 may perform N rotations to compute N bits for each index J _k . each iteration

In, the bit of index J _k

is the temporary angle

The value of k is compared to 0 and the associated value (0 or 1) is the bit of J _k

It can be determined by assigning to

의 값을 0과 비교할 수 있다. 만일 각도

가 0보다 작으면, LUT 인덱스 빌더(430)는 단계(546)로 계속 진행할 수 있고, 거기에서

에는 값 (-1)이 할당될 수 있으며, 각각의 인덱스 J_k의 비트

의 값에는 값 1이 할당될 수 있다. 만일 각도

가 0보다 크면, LUT 인덱스 빌더(430)는 단계(543)로 계속 진행할 수 있고, 거기에서

에는 값 1이 할당될 수 있으며, 각각의 인덱스 J_k의 비트

의 값에는 값 0이 할당될 수 있다. 단계 550에서, LUT 인덱스 빌더(430)는 각각의 인덱스 J_k의 비트

의 값을 업데이트하고, 임시 각도

에 가산될 수 있는 현재의 회전

을 계산할 수 있다(

를

로 승산함으로써). 그러면, LUT 인덱스 빌더(430)는 반복자

를 증분시켜서 각각의 인덱스 J_k의 다음 비트를 핸들링하게 할 수 있다.At step 530, the LUT index builder 430 calculates each angle

The value of can be compared to 0. if angle

If is less than zero, LUT index builder 430 may continue to step 546, where

A value (-1) may be assigned to, and the bits of each index J _k

A value of 1 may be assigned to the value of . if angle

If is greater than zero, LUT index builder 430 may continue to step 543, where

may be assigned a value of 1, and the bits of each index J _k

A value of 0 may be assigned to the value of . At step 550, the LUT index builder 430 uses the bits of each index J _k

update the value of , the temporary angle

turns of the current that can be added to

can be calculated (

cast

by multiplying by ). Then, the LUT index builder 430 is an iterator

It is possible to handle the next bit of each index J _k by incrementing .

에 대한 인덱스 J_k를 제공하고(단계(570)), 계산된 인덱스 J_k를 연상 메모리 어레이(410)의 열에 기록할 수 있다.At step 560, LUT index builder 430 may check whether all bits of J _k have been calculated. If no such index is ready, LUT Index builder 430 can return to step 520, and when that index is complete, LUT Index builder 430 outputs each input angle as output.

An index J _k for , may be provided (step 570), and the calculated index J _k may be written to a column of the associative memory array 410.

6, to which reference is now made, is a schematic diagram of a flow 600 implemented in accordance with one embodiment of the present invention and performed by a LUT index builder 440.

에 연관된 인덱스 J_k를 가지는 복수 개의 각도

를 수신할 수 있다.At step 601, the LUT value assigner 440 determines each angle

A plurality of angles with index J _k associated with

can receive

(LUT(420)의 엔트리들에 걸쳐서 반복하기 위하여 사용되며, 각각의 엔트리는 하나의 인덱스에 의하여 식별됨)를 0으로 초기화할 수 있다. 단계 620에서, LUT 값 할당기(440)는 최초의 반복을 연상 메모리 어레이(410)의 모든 열에 대해서 동시에 시작할 수 있다. 단계 630에서, 각각의 인덱스 J_k의 값이

에 비교될 수 있다. J_k의 값이

와 같으면, LUT 값 할당기(440)는 단계(640)로 계속 진행하고 LUT(420) 내의 엔트리

로부터의 X의 값을 각도

와 연관된 열에 복사하며, 단계(650)로 계속 진행할 수 있다. J_k의 값이

와 같지 않으면, LUT 값 할당기(440)는 단계(650)로 바로 진행할 수 있다. 단계 650에서 반복자

는 증분될 수 있고, 단계 660에서 LUT 값 할당기(440)는 흐름(600)이 LUT(420)의 마지막 엔트리에 도달했는지를 점검할 수 있다. 엔트리

가 마지막 엔트리가 아니면, LUT 값 할당기(440)는 단계(620)로 복귀하여 LUT(420)의 다음 엔트리를 핸들링할 수 있다. 그렇지 않으면, LUT 값 할당기(440)는 메모리 어레이(410)의 열에 기록된 각각의 각도

와 연관된

및

의 값들을 가지고서 종료할 수 있다(단계(670)).At step 610, the LUT value assigner 440 uses an iterator

(used to iterate through the entries of the LUT 420, each entry identified by an index) to zero. At step 620, the LUT value allocator 440 may start an initial iteration simultaneously for all columns of the associative memory array 410. In step 630, the value of each index J _k

can be compared to The value of J _k

, LUT value assigner 440 continues to step 640 and the entry in LUT 420

Angle the value of X from

copy to the column associated with , and may proceed to step 650 . The value of J _k

is not equal, LUT value assigner 440 may proceed directly to step 650 . Iterator at step 650

may be incremented, and at step 660 LUT value allocator 440 may check whether flow 600 has reached the last entry of LUT 420. entry

If is not the last entry, LUT value assigner 440 may return to step 620 to handle the next entry in LUT 420. Otherwise, the LUT value allocator 440 calculates each angle recorded in a column of the memory array 410.

associated with

and

may end with the values of (step 670).

7, to which reference is now made, is a schematic diagram of a flow 700 implemented in accordance with one embodiment of the present invention and performed by a CORDIC computer 450.

,

및

의 값 및 LUT 인덱스 빌더(430)에 의하여 계산된 임시 각도

의 값일 수 있다. CORDIC 알고리즘의 다음 M 회의 반복도 연상 메모리 어레이(410)의 모든 열에 대해서 동시에 실행될 수 있어서, 모든 입력 각도

에 대한 사인 및 코사인 값인 사인/코사인 추정기(400)의 계산의 최종 단계를 제공한다.The inputs to the CORDIC computer 450 at step 701 are all input angles, which are all stored in columns of the memory array 410.

,

and

The value of and the temporary angle calculated by the LUT index builder 430

can be the value of The next M iterations of the CORDIC algorithm can also be executed simultaneously for all columns of the associative memory array 410, so that all input angles

Provides the final step in the computation of the sine/cos estimator 400, which is the sine and cosine values for .

에 대하여 임시 파라미터

및

를 생성할 수 있고, 이들을 각각

및

로 초기화할 수 있다. 임시 파라미터는 CORDIC 알고리즘의 계산 전체에 걸쳐서 다음 반복에 대한 입력으로서 사용될 수 있다.At step 710, the CORDIC computer 450 determines each angle

About Temporary Parameters

and

can be generated, and each of them

and

can be initialized with Temporary parameters can be used as inputs to the next iteration throughout the computation of the CORDIC algorithm.

값은 0에 비교될 수 있다. 각도

가 0보다 작으면, 흐름(700)은

의 값이 (-1)로 설정될 수 있는 단계(736)로 계속 진행할 수 있다. 각도

가 0보다 크면, 흐름(700)은

의 값이 (+1)로 설정될 수 있는 단계(733)로 계속 진행할 수 있다.At step 720, each angle

Values can be compared to zero. Angle

If is less than zero, flow 700

Continuing to step 736 where the value of k may be set to (-1). Angle

If is greater than zero, flow 700 is

Continuing to step 733 where the value of k may be set to (+1).

에 대하여 동시에, CORDIC 알고리즘의 다음 반복을 계산할 수 있고, 현재의 반복

모두에 대한

및

의 값들을 수학식 7 및 8에서 규정된 바와 같은 임시 파라미터를 사용하여 업데이트할 수 있다:At step 740, the CORDIC computer 450 determines all angles.

At the same time, the next iteration of the CORDIC algorithm can be computed, and the current iteration

for all

and

The values of can be updated using temporary parameters as specified in Equations 7 and 8:

는

에 따라서 임시 각도

에 가산되거나 감산될 수 있고, 반복자

(미리 규정된 각도

에 걸쳐서 반복하기 위하여 사용됨)는 증분될 수 있다.current rotation angle

Is

Temporary angle according to

can be added to or subtracted from, and the iterator

(predefined angle

used to iterate over ) can be incremented.

-

및

-의 계산된 값이 각각 cos(

) 및 sin(

)의 값들을 제공할 수 있는 단계(760)로 계속 진행할 수 있다.At step 750, CORDIC computer 450 may check whether the last iteration has been completed. If the iteration is not the last iteration (i.e. iteration i is less than T) then the CORDIC computer 450 may return to step 710, and as checked at step 750 if the last iteration is to be performed the CORDIC computer (450) is each

-

and

The calculated value of - is respectively cos(

) and sin(

) may be provided to proceed to step 760 .

The number of iterations performed by the CORDIC computer 450 (e.g., M = 5) may be optimized to match the current performance of the hardware of the sine/cos estimator 400, but is not limited thereto, and future It can be appreciated that depending on the hardware capabilities may vary to obtain the best performance.

8, to which reference is now made, illustrates an associative memory device 410 used by sine/cos estimator 400. Associative memory array 410 includes a memory array 810 , a multi-row decoder 820 , a multi-column decoder 830 and a controller 840 .

Memory array 810 may be any suitable memory array, volatile or non-volatile, transient or non-transitory, and may include pure memory cells arranged in rows and columns. Cells in a column can be connected by bit line processors that can perform calculations in that column. Cells in a row can be connected by word lines that can activate cells in multiple columns. Data including inputs, intermediate results, and outputs may be stored in columns of memory array 810 .

Multi-row decoder 820 may be any suitable row decoder capable of activating multiple rows simultaneously. Multi-row decoder 820 can activate more than one row of memory array 810 at one time. When multiple rows are active, all columns of the memory array 810 can provide concurrent computation for the active rows when a read operation is performed and concurrent write operations when a write operation is performed. there is.

Multiple column decoder 830 includes any suitable column decoder capable of activating multiple columns simultaneously, and any suitable sensing circuitry capable of sensing a value on any bit-line connecting a column of cells. can include The multi-column decoder 830 may provide a result of a Boolean function performed between a plurality of cells of each column simultaneously activated by the multi-row decoder 820 . The multi-column decoder 830 can select which sensed columns to write back to the memory array 810 and can write values from multiple sense circuitry components simultaneously.

The controller 840 may control activation of the multi-row decoder 820 and the multi-column decoder 830 . The controller 840 can indicate to the multi-row decoder 820 which row to activate for the current operation, read or write, and also write the output from which column back into the memory array 810 in an optional write operation. whether or not, and rows in which data can be written can be displayed in the multi-column decoder 830.

The controller 840 may include various parts of the sine/cos estimator 400, such as a LUT index builder 430, a LUT value assigner 440, and a CORDIC computer 450.

It can be appreciated that the computation of the sine/cos estimator 400 may occur within an associative memory array as a result of multiple read and multiple write operations. Thus, the sine/cos estimator 400 can simultaneously implement any Boolean operation on all columns of the associative memory array 410, allowing the resulting extensively parallel computations to occur in place. (Each column can perform the necessary calculations for a single angle; activating multiple columns can result in simultaneous computation of trigonometric functions for multiple angles).

의 개수에 의존하지 않는다는 것이 이해될 수 있다.The computational complexity of the sine/cosine estimator 400 is the input angle

It can be understood that it does not depend on the number of .

의 계산을 연상 메모리 어레이(410)의 하나 이상의 전용 열 내에서 처리할 수 있다. 단일 각도의 사인 및 코사인을 계산하는 복잡도가 여러 각도의 사인 및 코사인을 계산하는 복잡도와 같다.The sine/cos estimator 400 may receive multiple angles as input, each angle

The computation of can be processed within one or more dedicated columns of the associative memory array 410. The complexity of computing the sines and cosines of a single angle is the same as the complexity of computing the sines and cosines of multiple angles.

에 대해서 동시에, 인덱스 J_k를 계산할 수 있다. 이러한 동작의 복잡도는 O(N)일 수 있고, N은 LUT 인덱스의 비트수이다.LUT index builder 430 for each angle

At the same time for , the index J _k can be calculated. The complexity of this operation may be O(N), where N is the number of bits of the LUT index.

For each entry in LUT 420, LUT value assigner 440 assigns the values of X and Y from entry J _k in LUT 420 to columns of associative memory array 410 that share the same index J _k . can be copied at the same time. The complexity of this operation is O(2 ^N ), where N is the number of bits of the LUT index.

For each angle at the same time, the CORDIC computer 450 can calculate the values of X and Y starting from iteration N for M additional iterations (the values for the first N iterations are taken from the LUT). The complexity of this operation is O(M), where M is the number of iterations to compute the CORDIC algorithm.

의 표현을 라디안 단위이고 점 뒤에 14 개의 비트가 있는 정규화된 부호 고정 소수점으로 수정할 수 있다는 것에도 주목할 수 있다. 정규화는 각각의 각도

(범위 [-π, π] 내에 속할 수 있음)를 π로 나누는 것을 포함할 수 있는데, 이것은 새로운 범위 [-1, 1]을 초래할 수 있다.In addition, the sine / cosine estimator 400 is an input angle

It may also be noted that we can modify the expression of to a normalized signed fixed-point in radians with 14 bits after the dot. Normalization is for each angle

(which may fall within the range [-π, π]) by π, which may result in a new range [-1, 1].

While standard CORDIC operation works with angles in the range of [-π/2, π/2], sine/cos estimator 400 operates over the full range of normalizable angles [-π, π]. can do. The sine/cos estimator 400 can transform the results, rearrange them to their original values before normalization, and normalize them again (divide by 2).

For 10 iterations, it can be understood that the minimum rotation angle is arctan(1/1024) radians, which is about 0.000976562 radians; Therefore, the maximum error of the sine/cos estimator 400 may be about 0.00097656203. (The maximum error is estimated by converting the maximum error in radians to a pure number (number without units). For T = 10, the minimum angle of rotation is 1/1024, so the maximum error for cosine is 4.768* 10^-7 and 0.00097656203 for the sine.)

The separate computation time of each method within the associative memory array 410 (10-bit index of the CORDIC algorithm or LUT with 10 iterations) is less than the computation time of the unified approach as described herein below. It is understandable that it is long.

For 10 iterations of the CORDIC algorithm: each iteration takes (68 + 2*i) where i is the number of iterations with a total of 770 cycles.

For a LUT covering 10 iterations (the size of the LUT is 2 ¹⁰ ): each iteration to compute the index J _k takes 24 cycles per iteration, for a total of 240 cycles.

The complexity of the lookup to send the value from LUT 420 to associative memory array 410 is 1024 iterations and 5 cycles per iteration, which add up to a maximum of 5k cycles.

The complexity of 10 iterations of the unified approach, where 5 iterations are covered by each part (LUT of size 2 ⁵ and 5 iterations of the CORDIC algorithm), is the flow using the CORDIC LUT for the first 5 iterations (300 ) of step 340 (FIG. 3) and step 350 of flow 300 using CORDIC calculations for the next five iterations.

Step 340 (LUT) consists of 5 iterations each taking 24 cycles to build the index J _k (total of 5 * 24 = 120 cycles), and to assign a value from the LUT to each angle. It contains 32 lookups each taking 5 cycles (5 * 32 = 180 in total), which makes 120 + 160 = 280 for the entire LUT operation.

The CORDIC part contains 5 iterations each taking 68 + 2*i cycles, which add up to less than 400 cycles.

Therefore, the total number of cycles for the unified approach is 280 + 400 = 680 cycles, which is less than using the LUT for 10 iterations (3000) or performing 10 iterations of the CORDIC algorithm (770). little.

The steps indicated for the flows herein are not intended to be limiting, and it can be appreciated that variations of each flow may be practiced. Such variations may include increasing the number of steps, reducing the number of steps, changing the sequence of steps, and omitting steps, among other variations that will be apparent to those skilled in the art.

Although specific features of the invention have been illustrated and described herein, many variations, permutations, alterations, and equivalents will now occur to those skilled in the art. Therefore, it should be understood that the appended claims are intended to cover all such modifications and variations as fall within the true spirit of this invention.

Claims (6)

As a method for an associative memory device,
providing a look-up table (LUT) with all possible solutions for the first N iterations of the CORDIC algorithm;
receiving a plurality of input angles;
simultaneously calculating a position index for each input angle of the plurality of input angles, and simultaneously storing each index in a row of the associative memory device;
copying a solution from the LUT at the location index into a plurality of columns associated with the index; and
concurrently performing M additional iterations of the CORDIC algorithm in the plurality of columns to compute a value of a trigonometric function for each of the plurality of input angles. According to claim 1,
Simultaneously calculating the position index,
Simultaneously for each of these input angles:
initializing a temporary angle as the value of the input angle;
determining a value and sign of a bit based on a comparison between the temporary angle and zero;
assigning the bit to the location index;
calculating a rotation angle based on the sign and a predetermined angle;
adding the rotation angle to the temporary angle; and
generating the location index of N bits by repeating the initializing, determining, assigning and calculating steps N times. According to claim 1,
A method for an associative memory device, where N = 5 and M = 5. According to claim 1,
The step of copying the solution is,
stepping sequentially through all entries of the LUT and simultaneously copying X values and Y values from location indices in the LUT to all columns having the same computed location index. As a sine and cosine estimator,
a look-up table (LUT) for storing previously calculated sine and cosine values for the first N iterations of the CORDIC algorithm;
an associative memory array for storing information related to a plurality of angles;
a LUT index builder for simultaneously constructing an index reflecting a rotation of a predefined rotation angle for each of the plurality of angle indices;
a LUT value allocator for assigning values from entries in the LUT to columns of the associative memory array sharing the same index; and
A CORDIC computer for providing sine and cosine values after N + M iterations of the CORDIC algorithm to the plurality of angles by simultaneously calculating M additional iterations of the CORDIC algorithm in the plurality of columns of the associative memory array. , sine and cosine estimators. According to claim 5
where N = 5, and the LUT contains 2 ⁵ entries.

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